growth of function given by infinite series

I apologize but I have a bit open-ended question... Imagine you have a sequence of real numbers that go to infinity very fast (let's say for any )

Then the series defines an analytic function on the whole complex plain except at 's where it has poles. I'm concerned with its behavior at infinity in the upper half plane. Is it possible to show that

It looks so obvious but I'm not sure how to show it. Am I being dumb? The second question is -- can we determine the leading term behavior? Is it always ?